A linear component in a system is often represented in frequency domain. To obtain the time-domain response of the system driven by sources with given waveforms, the spectrum of the linear component must be converted to time-domain impulse response function by inverse Fourier transform. The Fourier transform involves integration of the spectrum over frequencies from DC to infinity. The spectral data is often available only up to the highest frequency of measurement or electromagnetic (EM) simulation. Even in cases where the analytic model of the spectrum is known, the numerical integration of the Fourier transform has to be truncated at a finite frequency. One consequence of spectrum truncation is the violation of causality in the impulse response, namely non-zero response at negative time. To restore causality, adjustments have to be made in the impulse response, including shifting the response toward the positive side of the time or manually removing responses at negative time. Such adjustments cause discrepancy between the impulse response and the original spectrum, and therefore inaccurate time-domain response. A common technique to deal with spectrum truncation is to apply a low-pass window to the spectrum. However, windowing does not solve and can worsen causality problem as it is equivalent to convolution of a low-pass filter with the original physical response. This operation always leads to ripples or ringing in negative time. Thus, it is hard to control the spectrum accuracy when adjusting the impulse response in the time domain.
While any physical response must be causal with respect to t=0, e.g. zero response when t<0, response of transmission lines must also be causal with respect to delay, e.g. zero response when t<delay time. The delay time is the time it takes for the signal travels from the input end to the output end. Delay causality should be enforced in the transmission line impulse response calculation from band-limited spectrum.
Another issue related to causality is the passivity correction of S-parameter data. Due to measurement noise or numerical error in EM simulation, S-parameter data of a passive component can be slightly non-passive at certain frequencies. Passivity violation could lead to unstable and erroneous waveforms. In time-domain simulation, it is required that the corrected S-parameters remain to be causal.
The simplest way to correct passivity violation is to scale the whole spectrum by a constant factor which is less than one. This method will not affect the causality of the spectrum. However, it penalizes the non-passive frequencies but also the passive ones. Therefore, the corrected spectrum is overly damped when compared to the original data.